The Tories launched their own inquiry on Sunday after the resignation of Peter Cruddas, whose claims were filmed by undercover Sunday Times reporters.

Meanwhile, Labour demanded a separate, independent, inquiry to investigate the suspicion that Tory donors can gain secret access to Cameron, the chancellor, George Osborne, and their advisers. Cross-party talks on the funding of political parties, which were due to start in a few weeks, have been brought forward to this week.

Police were also asked on Sunday to investigate claims that the law on political donations may have been broken. According to the Sunday Times, Cruddas believed any prospective donations from the reporters would come from Liechtenstein and would be ineligible under election law.

He was reported to have discussed the creation of a British subsidiary and the possibility of using UK employees to make the donation. A campaigner, Mark Adams, said he had asked police to investigate. "The article indicates that the party has done this before," he said.

The revelations reinforce the impression of a close relationship between Cameron's Tories and millionaire businessmen. They come hard on the heels of the storm over last week's budget, which has been condemned across the political spectrum for giving the rich a cut in the top rate of income tax while penalising pensioners.

Cruddas, a generous Tory donor who set up the betting company CMC Markets, was appointed last June as co-treasurer and became treasurer this month. He told the undercover reporters that they should pay £250,000 to gain "premier league" access to Cameron, Osborne and No 10's policy team.

"Two hundred grand to 250 is premier league … what you would get is, when we talk about your donations, the first thing we want to do is get you at the Cameron/Osborne dinners," he said.

This could lead to access to dinners at Cameron's private apartment in Downing Street, Cruddas told the reporters, who were posing as representatives of a fictitious wealth fund.

Once inside, they could ask Cameron "practically any question you want".

"If you're unhappy about something, we will listen to you and put it into the policy committee at No 10. We feed all feedback to the policy committee," he said.

Cameron, in an attempt to quell the row, said: "This is not the way we raise money in the Conservative party. It shouldn't have happened. It's quite right that Peter Cruddas has resigned. I will make sure there is a proper party inquiry to make sure this can't happen again."

Labour called for an independent inquiry into Cruddas's comments as well as a list of the Tory donors who have visited government property and of those who have made submissions to the Downing Street policy unit. While details of official meetings between the prime minister and donors at No 10 and Chequers are publicly listed, private dinners are not.

The Labour leader, Ed Miliband, told the BBC that a Tory inquiry into what had happened was not good enough. "These allegations can't be swept under the carpet – there needs to be a proper independent investigation into what influence was sought, what influence was gained and what impact it had," he said.

Miliband said it "would be right" for Cameron to make a statement to parliament on the issue. In particular, he will be expected to explain which donors have had access to his private office.

A Downing Street source said private meetings and social engagements are not published and made clear that Cruddas had never been to Cameron's Downing Street flat.

In his resignation statement, Cruddas said: "I deeply regret any impression of impropriety arising from my bluster in that conversation. Clearly there is no question of donors being able to influence policy or gain undue access to politicians.

"Specifically, it was categorically not the case that I could offer, or that David Cameron would consider, any access as a result of a donation. Similarly, I have never knowingly even met anyone from the No 10 policy unit."

Cruddas was last night replaced as treasurer by Lord Fink, a hedge fund manager and former CEO of Man Group. Senior Conservatives were last night attempting to portray Cruddas as boastful and naive. Sources said that he was "bullshitting" and "trying to impress" and described his claims of influencing government policy as "ludicrous". Cruddas was put under pressure to resign as soon as the full facts became known on Saturday night, one insider said.

Conservative Party deputy chairman Michael Fallon told Sky News that Mr Cruddas had been "blustering and boasting and that was clearly wrong". A Tory party spokesman claimed on Sunday afternoon that Andrew Feldman, the party's chairman and a close friend of Cameron who appointed Cruddas to his position, would be involved in the internal inquiry into Cruddas's comments. Later, the spokesman said Feldman may not be involved in the inquiry after all. The three major parties have put forward two representatives each for the cross-party talks on funding reform which are now expected to begin this week, sources said on Sunday. Andrew Feldman, the Tory chairman, will represent the Tories alongside Francis Maude.

During a review of party funding by the Committee on Standards in Public Life last year, the Tories suggested an annual cap of £50,000 per donor.

This was dismissed by the committee as it amounted to twice the average UK salary and over a five-year parliament added up to £250,000. Instead, a cap of £10,000 per donor was recommended by the committee, with an extra £23m of taxpayers' money, to be shared between parties.

Committee chairman Sir Christopher Kelly said: "The only way to remove the suspicion surrounding very large donations would be to ban very large donations. For the Conservatives it means giving up their advantage of having more wealthy supporters, for Labour, the relationship with the trade unions," he said.